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Chapter 11: Problem 38

Show that \(\vec{A} \cdot(\vec{A} \times \vec{B})=0\) for any vectors \(\vec{A}\) and \(\vec{B}\).

Short Answer

Expert verified
The identity \(\vec{A} \cdot(\vec{A} \times \vec{B})=0\) is proven to be true for any vectors A and B.

Step by step solution

01

Express the Given Identity

The given identity is \(\vec{A} \cdot(\vec{A} \times \vec{B})=0\). This is saying that the dot product of vector A with the cross product of A and B is equal to zero.
02

Apply the Distributive Property

First apply the distributive property of the dot product: \(\vec{A} \cdot (\vec{A} \times \vec{B}) = \vec{A} \cdot \vec{A} \times \vec{B} - \vec{A} \cdot \vec{B} \times \vec{A}\).
03

Use the Anticommutative Property

The anticommutative property of the cross product states that for any two vectors U and V, U x V = - V x U. So, \(\vec{A} \cdot \vec{A} \times \vec{B} - \vec{A} \cdot \vec{B} \times \vec{A} = \vec{A} \cdot \vec{A} \times \vec{B} + \vec{A} \cdot \vec{A} \times \vec{B}\).
04

Simplify

After simplifying we get \(2 \vec{A} \cdot \vec{A} \times \vec{B}\). Since any vector cross itself is zero, we get the final result as \(0=0\).

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